WO2010049564A2

WO2010049564A2 - Optimized-cost method for computer-assisted calculation of the aerodynamic forces in an aircraft

Info

Publication number: WO2010049564A2
Application number: PCT/ES2009/070464
Authority: WO
Inventors: Angel Gerardo VELÁZQUEZ LÓPEZ; Diego ALONSO FERNÁNDEZ; José Manuel VEGA DE PRADA; Luis Santiago Lorente Manzanares
Original assignee: Airbus Operations, S.L.; Universidad Politécnica de Madrid
Priority date: 2008-10-28
Filing date: 2009-10-28
Publication date: 2010-05-06
Also published as: RU2510969C2; US20100106471A1; RU2011121577A; CA2741655C; CA2741655A1; US8041547B2; EP2360606B1; EP2360606A2; WO2010049564A3; ES2595979T3; BRPI0921804A2

Abstract

Optimized-cost method for computer-assisted calculation of the aerodynamic forces in an aircraft, so as to provide values of dimensional variables, dependent on a predefined set of parameters, for the entire aircraft or aircraft component, which comprises the following steps: a) defining a mesh in the parametric space; b) obtaining a suitable reduced order model (ROM), in particular a POD model, for calculating said variables for any point in the parametric space by means of a repetitive process. CFD is used to calculate said variables at a set of suitably selected points in the parametric space, which are used to obtain approximate values, via ROM and ad-hoc interpolation, of the dimensional variables at any other point of the parametric space. The method minimizes the required number of CFD calculations (so as to minimize the computational cost which depends basically on this number) for a given error level.

Description

METHOD OF CALCULATION ASSISTED BY COMPUTER OF AERODYNAMIC FORCES IN A COST OPTIMIZED AIRCRAFT

FIELD OF THE INVENTION

The present invention relates to methods of assistance for the design of aircraft by making optimized calculations of the aerodynamic forces experienced by the entire aircraft or by a component of the aircraft.

BACKGROUND OF THE INVENTION

A common situation in industrial practical applications related to product development is the need to carry out rapid analyzes in a state parameter space. In the specific case of the aeronautical industry, the calculation of the aerodynamic forces experienced by an aircraft is an important element for the optimal design of its structural components so that the weight of the structure is the minimum possible, being able to the same of time to resist the expected aerodynamic forces.

Thanks to the increase in the use of Computer Fluid Simulation, the determination of aerodynamic forces in an aircraft is usually done today by numerically solving the average Reynolds equations of Navier-Stokes (RANS equations onwards) that model the movement of the flow around the aircraft, using discrete finite element models or finite volume models. With the demand for accuracy required in the aeronautical industry, each of these calculations requires significant computational resources.

The aerodynamic sizing forces are not known a priori and how the overall magnitude of the forces can depend on many different flight parameters such as angle of attack, angle of sliding, Mach number, deflection angle of the control surface, it has been necessary to carry out long and expensive calculations to properly calculate the maximum aerodynamic forces experienced by the different components of an aircraft or by the entire aircraft. In order to reduce the global number of these long calculations, approximate mathematical modeling techniques have been developed in the past to obtain a Reduced Order Model (ROM) such as the Decomposition in Singular Values (SVD) as a means to carry out interpolations intelligent, or the most exact Orthogonal Decomposition of Covariance (POD onwards) that takes into account the physics of the problem by using a Galerkin projection of the Navier-Sto kes equations.

The idea of these techniques is to define the new analytical solution as a combination of the information obtained previously. POD defines several modes that include the solution obtained by Computational Fluid Dynamics (CFD) and then uses those modes to reproduce solutions not obtained by CFD. The application of these techniques may require many CFD calculations, which entails a large computational cost.

The present invention is oriented to the solution of this drawback.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide methods for making analytical calculations of the aerodynamic forces experienced by an entire aircraft or by a component of the aircraft, said forces being dependent on a significant number of parameters, minimizing the computational cost.

It is another object of the present invention to provide methods for making analytical calculations of the aerodynamic forces experienced by an entire aircraft or by a component of the aircraft, said forces being dependent on a significant number of parameters, minimizing the number of CFD calculations.

These and other objects are achieved by an appropriate computer-assisted method to assist in the design of an aircraft by providing the values of one or more dimensional variables, such as

The pressure distribution on the surface of a wing, for the entire aircraft or for a component of the aircraft, said one or more dimensional variables being dependent on a predefined set of parameters, such as a set that includes the angle of attack and Mach number, which includes the following steps:

- Define a mesh in the parametric space by establishing predetermined distances between their values.

- Obtain an appropriate model to calculate said one or more dimensional variables for any point of the parametric space through an iterative process with respect to a small group of points, of increasing number of members in each iteration, comprising the following sub-steps:

• Calculate the values of said one or more dimensional variables for an initial group of points using a CFD model.

• Obtain a ROM model from said CFD calculations and calculate the values of said one or more dimensional variables for said initial group of points using the initial ROM model.

• Select the e-point of the group with the maximum deviation ε between the results provided by the CFD and ROM models and finish the iterative process if ε is less than a predefined value ε ₀ .

• Select as new points in the parametric space to be added to the group of points those points placed at a predetermined distance from said e-point in the mesh of the parametric space. - TO -

• Calculate the values of said one or more dimensional variables for the new points using the CFD and ROM models and repeat the process from the third sub-step.

In particular, said one or more dimensional variables include one or more of the following: aerodynamic forces, the values in the skin and the distribution of values around the entire aircraft or a component of

The aircraft; said set of parameters includes one or more of the following: angle of attack and Mach number; and said aircraft component is one of the following: a wing, a horizontal tail stabilizer, a vertical tail stabilizer.

In a preferred embodiment, said entire aircraft or aircraft component is divided into blocks and said CFD and ROM models are applied block by block. With this, a very exact method is obtained to provide the values of one or more dimensional variables of an entire aircraft or of a component of the aircraft.

In another preferred embodiment said ROM model is a POD model. CFD is used to calculate the distribution of pressure at a set of points in the parametric space, appropriately selected, which are used to obtain approximate values, via POD and ad-hoc interpolation, of the dimensional variables at any other point in the parametric space. Additionally, the method minimizes the required number of CFD calculations (to minimize the computational cost, which basically depends on that number) for a given level of error. This is done using POD and interpolation with previously calculated points. New points are selected iteratively either one by one or in groups. A method is thus obtained to provide the values of one or more dimensional variables of an entire aircraft or of a component of the aircraft, dependent on a predefined set of parameters, optimizing computational costs.

Other features and advantages of the present invention will become apparent from the following detailed description of the embodiments, illustrative of its object, together with the attached figures. DESCRIPTION OF THE FIGURES

Figure 1 shows views of the suction side, the pressure side, the leading edge and the tip of a wing of an aircraft divided into blocks.

Figure 2 shows a graphical representation of a local sub-mesh of the mesh of the parametric space to select new points to add to the group of points that is used to obtain the POD model according to this invention.

DETAILED DESCRIPTION OF THE INVENTION

We will now describe a method according to the present invention to obtain a POD model that allows to calculate the distribution of stationary pressure on the surface of the wing of an aircraft, said pressure distribution being dependent on two free parameters: angle of attack (α) and Mach number (M). Initial Steps:

Step 1: Division of the wing into several blocks according to the geometry of the object. CFD tools usually divide the 3D computational domain into blocks as illustrated in Figure 1, which shows the wing divided into 16 main blocks. This is a convenient, but not essential part of the method, which can be applied using a single block.

Step 2: A definition of the mesh of the parametric space is carried out by establishing an initial value of the minimum distance for each parameter in the parametric space di, I = 1, ..., parameter # based on a first conjecture of the Ia minimum distance between points in the parametric space and that may need some calibration in subsequent steps. That distance will be reduced, if necessary, by the method during the iteration. In that case an equiespaced mesh is defined in the parametric space based on those distances That mesh will evolve during the process and may become unquiespaced.

For example, if the parameters considered are angle of attack (α), in the range -3 ^or + 3 °, and Mach number (M), in the range 0.40 to 0.80, the mesh of the parametric space can be defined by establishing the distances d _α = 0.5 and d _M = 0.05.

Step 3: Initiation of the process for an initial group of points in the parametric space selected by the user such as the following:

Group

Initial Mach Alfa

P1 0.400 -3.00

P2 0.600 -3.00

P3 0.800 -3.00

P4 0.400 0.00

P5 0.600 0.00

P6 0.800 0.00

P7 0.400 3.00

P8 0.600 3.00

P9 0.800 3.00

Introduction of a new group of points

Step 4: Application block by block of POD to the initial group of points. A block-dependent set of modes is obtained for each block:

P where P is the pressure distribution, X ₁ is the spatial coordinates, oc is the angle of attack, M is the Mach number, A _p are the amplitudes of the modes, and the columns of the matrix φ ^ are the POD modes . Each mode has an associated singular value that results from the application of POD. Step 5: Classification of modes: S The first classification that is made (in each block) in two parts is the following: (a) those modes that give an RMSE less than a preset value E ₁ (dependent on ε ₀ after some calibration) are neglected; (b) the nor modes that are retained are called principal modes.

/ In turn, the main modes are classified into these two groups: n primary modes and n _x -n secondary modes, obtaining n after some calibration, such as

4 n = -n _γ .

The mean square error (RMSE) is defined as:

where N _p is the total number of points of the mesh that defines the wing, and er ^ is the difference between the pressure of the approach and the pressure of the CFD solution at the point / of the mesh. Step 6: POD reconstruction of the pressure distribution for each of the already calculated groups of points using the primary (n) modes in each block. Then, an additional approximation is carried out for each point using the neighboring points via least squares.

Step 7: Comparison between the pressure profiles resulting from the CFD calculation and the POD + interpolation and estimation of the RMSE in each block at each of the points already calculated.

The RMSE for the initial nine point group mentioned above is as follows:

RMSE

P1 0.0371

P2 0.0298

P3 0.0887 P4 0.0273

P5 0.0190

P6 0.0756

P7 0.0605

P8 0.0930

P9 0.1758

Step 8: Select the point with the highest RMSE. As shown in the table above in the first iteration that point is P9.

Step 9: Definition, as illustrated in Figure 2, of a local sub-mesh of the entire parametric space mesh in the area near point 21 of maximum error. This local sub-mesh has three levels at distances d _¡ (first level), 2 - d _¡ (second level) and 4 - d _¡ (third level).

Step 10: Selection of the level at which the new point will be introduced. If there is any point between two levels (see below) it is considered that they belong to the lower level.

• If there is no point present in the entire sub-mesh, the new point is introduced in the third level.

• If there are only points in the third level, then the new point is introduced in the second level. • If there are no points in the first level and there is only one point in the second level, the new point is introduced in the second level.

• If there are no points in the first level and there are at least two points in the second level, the new point is introduced in the first level.

• If there is at least one point in the first level, then the new point is introduced in the first level with an exception that implies the introduction of a sub-level in the local mesh. This occurs when (a) there are at least five points in the first level, and (b) at least four of those points show the highest RMSE among all points in the three levels. In this case, the distances in the local sub-mesh are divided by two and step 9 is repeated again with the new sub-mesh resulting. It should be noted that this step means that each point will generally have a set of minimum distances d different.

In the example we are considering, the new point P10 is introduced in the third level because none of the initial group points is present in the local sub-mesh in the environment of P9.

Step 11: Once the target level has been chosen, the point of greatest space covering is selected as follows. The minimum distance, D, of each possible candidate to the remaining points of the group already selected is calculated. The candidate with the highest value of D is selected. D is the distance in the parametric space. In this example, the distance between two points of the parametric space (labeled 1 and 2) is defined as follows:

where OC ₁₉ = - and M ₁₉ = - are the distances in the oc parameters

¹² Aa ^or AM and M, and Δα and AM the corresponding total ranges of those parameters. In the example we are considering the distance between points of the third level and the closest point belonging to the group is shown in the following table.

Third point closest point group level Distance

Mach Alfa Mach Alfa

0.650 3.00 0.600 3.00 0.1250

0.650 2.50 0.600 3.00 0.1502

0.650 2.00 0.600 3.00 0.2083

0.650 1.50 0.600 0.0 0.2795

0.700 1.50 0.600 0.0 0.3536

0.750 1.50 0.800 0.0 0.2795

0.800 1.50 0.800 0.0 0.2500 Therefore, the new point that is introduced is P10: Mach = 0.700, Alpha = 1.50.

Step 12: If more than one point is entered in each iteration, then the process is repeated from step 8 excluding the points already selected. Mode set update: Once the new point (or group of points) has been calculated, the set of modes for each block is updated.

Step 13: Application of POD to the group of points, ignoring those modes that have an RMSE less than E ₁ . Step 14: Calculation of some pseudo-points, defined block by block, that form two groups:

• The n _x main modes of each block multiplied by their respective singular values.

• The POD modes obtained by applying POD to the new points resulting from the last iteration, multiplied by their respective singular values.

Steps 13 and 14 can be recast in one step. In this case, the pseudo-points are defined by adding together the main modes of the points already calculated, multiplied by their respective singular values, and the new points. The division in the mentioned steps 13 and 14 is done to filter numerical errors of the process which is a well known advantage of the POD method.

Step 15: POD application to the set of all pseudo-points, block by block. Step 16: Repeat the process from step 5.

To illustrate this iterative process, a brief description of the second iteration follows in the example we are considering:

The RMSE for the group of 10 points in the second iteration is as follows:

RMSE

P1 0.0313

P2 0.0242

P3 0.0723

P4 0.0275

P5 0.0167

P6 0.0569

P7 0.0853

P8 0.0458

P9 0.1421 P10 0.0260

The point with the maximum error is still P9 and the new point P11 is introduced in the second level because there is no point in the group at levels 1 and 2 and there is a point at level 3 (P10 introduced in the first iteration). The distance between the second level points and the nearest point belonging to the group is shown in the following table:

Second points Nearest point group level Distance

Mach Alfa Mach Alfa

0.700 3.00 0.800 3.00 0.2500

0.700 2.50 0.700 1.50 0.1667

0.700 2.00 0.700 1.50 0.0833

0.750 2.00 0.700 1.50 0.1502

0.750 2.00 0.800 3.00 0.1662

Therefore the new point that is introduced is P11: Mach = 0.700, Alpha = 2.50. Stop Criteria:

Step 17: The process is terminated when the RMSE, calculated in Step 7 using POD and linear and least squares interpolation is less, in both cases than ε ₀ .

Results

In the execution of the method in the example we are considering the initial set of points was, as already said, the following:

Mach Alfa

P1 0.400 -3.00

P2 0.600 -3.00

P3 0.800 -3.00

P4 0.400 0.00

P5 0.600 0.00

P6 0.800 0.00

P7 0.400 3.00 P8 0.600 3.00 P9 0.800 3.00

Throughout the iterative process the following points were added to the group:

P10 0.700 1.50

P11 0.700 2.50

P12 0.800 2.00

P13 0.500 1.50

P14 0.750 2.50

P15 0.400 2.00

P16 0.700 -1.00

P17 0.750 1.50

P18 0.750 3.00

P19 0.800 -1.50

P20 0.500 2.50

P21 0.800 2.50

P22 0.800 1.50

P23 0.700 0.50

P24 0.750 1.00

P25 0.700 3.00

P26 0.750 2.00

P27 0.450 2.50

P28 0.800 1.00

P29 0.450 3.00

P30 0.750 -0.50

An evaluation of the model obtained according to the method of this invention can be made by comparing the results obtained in 16 points using said model in several iterations and using the CFD model shown in the following table:

Results Coefficient of the Model of the Invention PFD support

Points . _« . 10 15 20 25 30. _« . Mach Alfa Shows Points Points Points Points Points

Tp1 0.800 2.25 0.1965 0.1922 0.1966 0.1965 0.1971 0.1966

Tp2 0.800 1.25 0.1045 0.1061 0.1082 0.1075 0.1054 0.1058

Tp3 0.800 -1.25 -0.1077 -0.1089 -0.1085 -0.1073 -0.1082 -0.1088

Tp4 0.800 -2.25 -0.1920 -0.1871 -0.1925 -0.1927 -0.1928 -0.1936 Tp5 0.775 2.25 0.1895 0.1899 0.1899 0.1903 0.1910 0.1900

Tp6 0.775 1.25 0.1012 0.1036 0.1051 0.1031 0.1023 0.1018

Tp7 0.775 -1.25 -0.1048 -0.1018 -0.1121 -0.1057 -0.1066 -0.1068

Tp8 0.775 -2.25 -0.1867 -0.1853 -0.1884 -0.1908 -0.1912 -0.1916

Tp9 0.725 2.25 0.1773 0.1849 0.1778 0.1788 0.1777 0.1774

Tp10 0.725 1.25 0.0966 0.0971 0.0980 0.0965 0.0970 0.0970

Tp11 0.725 -1.25 -0.1002 -0.0962 -0.1078 -0.1022 -0.1029 -0.1022

Tp12 0.725 -2.25 -0.1785 -0.1812 -0.1816 -0.1829 -0.1867 -0.1864

Tp13 0.525 2.25 0.1577 0.1565 0.1267 0.1563 0.1561 0.1585

Tp14 0.525 1.25 0.0868 0.0722 0.0845 0.0847 0.0873 0.0854

TP15 0.525 -1.25 -0.0897 -0.0749 -0.0960 -0.0786 -0.0964 -0.1084

TP16 0.525 -2.25 -0.1600 -0.1580 -0.1598 -0.1196 -0.1199 -0.1197

Moment X Coefficient Results of the Model of the Invention

CFD points

Mach Alfa 10 15 20 25 30 Sample Points Points Points Points Points

Tp1 0.800 2.25 +0.2062 0.1979 0.2054 0.2054 0.2068 0.2061

Tp2 0.800 1.25 +0.1109 0.1124 0.1181 0.1174 0.1128 0.1127

Tp3 0.800 -1.25 -0.1018 -0.1023 -0.1024 -0.1010 -0.1016 -0.1022

Tp4 0.800 -2.25 -0.1866 -0.1810 -0.1867 -0.1866 -0.1866 -0.1870

Tp5 0.775 2.25 +0.1991 0.1957 0.1984 0.1992 0.2010 0.1995

Tp6 0.775 1.25 +0.1078 0.1102 0.1140 0.1117 0.1090 0.1085

Tp7 0.775 -1.25 -0.0987 -0.0953 -0.1067 -0.0993 -0.0999 -0.1000

Tp8 0.775 -2.25 -0.1812 -0.1790 -0.1824 -0.1846 -0.1848 -0.1850

Tp9 0.725 2.25 +0.1849 0.1910 0.1858 0.1875 0.1853 0.1849

Tp10 0.725 1.25 +0.1036 0.1041 0.1060 0.1029 0.1036 0.1037

Tp11 0.725 -1.25 -0.0939 -0.0894 -0.1018 -0.0955 -0.0959 -0.0954

Tp12 0.725 -2.25 -0.1728 -0.1746 -0.1749 -0.1760 -0.1798 -0.1796

Tp13 0.525 2.25 +0.1654 0.1644 0.1279 0.1637 0.1637 0.1658

Tp14 0.525 1.25 +0.0943 0.0809 0.0926 0.0928 0.0953 0.0933

TP15 0.525 -1.25 -0.0827 -0.0668 -0.0879 -0.0704 -0.0879 -0.1001

Tp16 0.525 -2.25 -0.1534 -0.1499 -0.1514 -0.1100 -0.1100 -0.1096

Coefficient Moment and Results of the Model of the Invention

CFD points

Mach Alfa 10 15 20 25 30 Sample Points Points Points Points Points

Tp1 0.800 2.25 -0.1068 -0.1044 -0.1076 -0.1074 -0.1081 -0.1076

Tp2 0.800 1.25 -0.0345 -0.0377 -0.0392 -0.0387 -0.0361 -0.0363

Tp3 0.800 -1.25 +0.1270 0.1278 0.1279 0.1266 0.1273 0.1278

Tp4 0.800 -2.25 +0.1914 0.1877 0.1921 0.1921 0.1923 0.1928 Tp5 0.775 2.25 -0.1036 -0.1036 -0.1038 -0.1044 -0.1054 -0.1043

Tp6 0.775 1.25 -0.0340 -0.0374 -0.0384 -0.0367 -0.0351 -0.0347

Tp7 0.775 -1.25 +0.1232 0.1215 0.1295 0.1241 0.1247 0.1248

Tp8 0.775 -2.25 +0.1858 0.1853 0.1878 0.1892 0.1896 0.1898

Tp9 0.725 2.25 -0.0960 -0.1017 -0.0970 -0.0982 -0.0967 -0.0965

Tp10 0.725 1.25 -0.0335 -0.0344 -0.0356 -0.0337 -0.0338 -0.0338

Tp11 0.725 -1.25 +0.1171 0.1151 0.1241 0.1188 0.1193 0.1188

Tp12 0.725 -2.25 +0.1770 0.1800 0.1805 0.1807 0.1833 0.1831

TP13 0.525 2.25 -0.0868 -0.0877 -0.0618 -0.0849 -0.0847 -0.0867

TP14 0.525 1.25 -0.0321 -0.0233 -0.0302 -0.0302 -0.0321 -0.0307

Tp15 0.525 -1.25 +0.1029 0.0911 0.1067 0.0924 0.1078 0.1172

Tp16 0.525 -2.25 +0.1564 0.1542 0.1548 0.1219 0.1221 0.1218

The modifications that fall within the scope of the following claims can be introduced in the preferred embodiment that we have described.

Claims

1.- An appropriate computer-assisted method to assist in the design of an aircraft by providing the values of one or more dimensional variables for the entire aircraft or for a component of the aircraft, said one or more dimensional variables being dependent on a predefined set of parameters, characterized in that it comprises the following steps: a) Define a mesh in the parametric space by establishing predetermined distances between its values; b) Obtain an appropriate model to calculate said one or more dimensional variables for any point of the parametric space through an iterative process with respect to a small group of points, of increasing number of members in each iteration, comprising the following sub-steps: b1 ) Calculate the values of said one or more dimensional variables for an initial group of points using a CFD model; b2) Obtain a ROM model from said CFD calculations and calculate the values of said one or more dimensional variables for said initial group of points using the initial ROM model; b3) Select the e-point of the group with the maximum deviation ε between the results provided by the CFD and ROM models and finish the iterative process if ε is less than a predefined value ε ₀ , b4) Select as new points in the parametric space to be added to the group of points those points placed at a predetermined distance from said e-point in the mesh of the parametric space; b5) Calculate the values of said one or more dimensional variables for the new points using the CFD and ROM models and repeat the process from sub-step b3).

2. A computer-assisted method according to claim 1, characterized in that said entire aircraft or aircraft component is divided into blocks and said CFD and ROM models are applied block by block.

3. A computer-assisted method according to any of claims 1-2, characterized in that said one or more dimensional variables include one or more of the following: aerodynamic forces, values in the skin, distribution of values around the entire aircraft or of a component of the aircraft.

4. A computer-assisted method according to any of claims 1 -3, characterized in that said set of parameters includes one or more of the following: angle of attack, Mach number.

5. A computer-assisted method according to any of claims 1 -4, characterized in that said component of the aircraft is one of the following: a wing, a horizontal tail stabilizer, a vertical tail stabilizer.

6. A computer-assisted method according to any one of claims 1-5, characterized in that said ROM model is a POD model.

7. A computer-assisted method according to claim 6, characterized in that the deviation ε between the results provided by the CFD and POD models is obtained as the mean square error between said results.

8. A computer-assisted method according to any of claims 6-7, characterized in that the POD model is obtained by eliminating the less relevant modes from the group of points.